Optimizing thermoelectric performance of carbon-doped h-BN monolayers through tuning carrier concentrations and magnetic field

The thermoelectric properties of carbon-doped monolayer hexagonal boron nitride (h-BN) are studied using a tight-binding model employing Green function approach and the Kubo formalism. Accurate tight-binding parameters are obtained to achieve excellent fitting with Density Functional Theory results for doped h-BN structures with impurity type and concentration. The influence of carbon doping on the electronic properties, electrical conductivity, and heat capacity of h-BN is studied, especially under an applied magnetic field. Electronic properties are significantly altered by doping type, concentration, and magnetic field due to subband splitting, merging of adjacent subbands, and band gap reduction. These modifications influence the number, location, and magnitude of DOS peaks, generating extra peaks inside the band gap region. Heat capacity displays pronounced dependence on both magnetic field and impurity concentration, exhibiting higher intensity at lower dopant levels. Electrical conductivity is increased by double carbon doping compared to single doping, but is reduced at high magnetic fields because of high carrier scattering. The electronic figure of merit ZT increases with lower impurity concentration and is higher for CB versus CN doping at a given field strength. The power factor can be improved by increasing magnetic field and decreasing doping concentration. In summary, controlling doping and magnetic field demonstrates the ability to effectively engineer the thermoelectric properties of monolayer h-BN.


Theoretical tight binding formalism
The Hamiltonian matrix should be calculated to obtain the band structure of the h-BN structure.For the pure h-BN case, the unit cell contains two atoms and the Hamiltonian matrix is described with the 2 × 2 matrix 63 .When impurities are added to the structure, controlling the impurity concentrations is essential and can be defined based on the ratio of impurity atoms to the all atoms.So, by increasing the number of unit cells in the structure, different concentrations of impurities can be modeled.In order to model different impurity concentrations, the pristine h-BN unit cell can be extended along the monolayer's primitive lattice vectors a1 and a 2 to create a supercell of dimensions (Na 1 ,Na 2 ), where N is the number of unit cells in each direction.The supercell of pure h-BN structure with N unit cells has N 2 boron type and N 2 nitrogen type atoms.For a supercell containing www.nature.com/scientificreports/m impurity atoms, the impurity concentration is given by m 2N 2 .By increasing the supercell size N, different impurity concentrations can be achieved while keeping the number of impurities m fixed.In this study, the structure with one and two impurity atoms and three (N 3 ), four (N 4 ) and five (N 5 ) unit cells have been selected and these supercells correspond to impurity concentration lies in between 2 and 5% for single impurity and 4-10% for double impurity, respectively.The schematic views of supercells with N = 4 are shown in Fig. 1 and the required tight binding parameters for these structures are obtained from DFT calculations [Supporting file].For the selected structures, the total Hamiltonian in second quantization is given by: where the first and second parts are correspond to the local onsite energies and interaction between neighbor atoms, respectively.The C †σ i,α and C σ i,α represent the creation and annihilation operators with spin σ for α-th atom type in i-th unit cell.The onsite energy for α-th type with spin σ in the primitive unit cell i , is shown with the ε σ i (α).
Calculating the Hamiltonian matrix required specifying tight binding parameters such as the hopping integrals between neighboring atoms and the on-site energies for each atom type.The relevant atom types are boron and nitrogen, which form the pristine h-BN structure, and carbon atoms as impurities.
DFT calculations were performed to obtain the band structure of carbon-doped h-BN structures with different impurity concentrations (Fig. S1).By fitting the tight binding band structure to the corresponding DFT results, the required tight binding hopping integrals and on-site energies were extracted for each atom type.This obtained tight binding parameters that can reproduce the DFT band structure of carbon-doped h-BN using the values given in Table 1.
By applying the magnetic field B T = σ gµ B B 0 , the Hamiltonian matrix becomes spin dependent and it labeled by H σ (k) for the spin up and down.The calculations in this work utilize a rescaled magnetic field value of B T = σ gµ B B 0 .The matrix form for Hamiltonian H(k) in the k space can be obtained with the Fourier transfor- mation for the creation and annihilation fermion operators as: The BN doped structure with N unit cell, has 4N 2 × 4N 2 Hamiltonian matrix and its band structure E σ (k) can be obtained by solving the Schrodinger equation where the wave vector k ≡ k x , k y is surrounded in the first Brillouin zone.
(1) Table 1.Tight binding parameters for carbon-doped h-BN.The table provides the optimized tight binding parameters (in eV), including on-site energies and hopping integrals between different atom pairs.These parameters were obtained by fitting the tight binding model to DFT band structures.On-site energy (eV) where ω n = (n+1)π k B T is the fermionic Matsubara frequencies, t il is the hopping integral matrix and (B T ) is the on-site energy matrixe in the presence of magnetic field, respectively.The DOS is obtained from imaginary part of Green's function with −1 π Im[G lj (E)].The thermal properties of doped graphene are calculated using the Kubo-Greenwood formula, and this formalism requires calculating the spectral function A(k, ε) = −2Im Ĝ(k, ε) which is related to the DOS.In the presence of the temperature gradient ∇T , the electrical charge current J e and thermal heat current J Q are defined in terms of the transport coefficients 64 and the ϒ mn coefficients are obtained in terms of the correlation function between J e and J Q current operators as 65 : The charge current operator function Ĵe and the heat current operator ĴQ can be defined as 64   where ν (p) is the velocity operator for the p-th Hamiltonian eigenvalues in presence of the mag- netic field.By using the Wick's theorem based on the Green function 0) , the cor- relation function parts can be written as: By using the Fourier transform of Green function G p (k, τ ) = m e −iω m τ G p (k, iω m ) , the transport coefficients Eq. ( 4) can be expressed as: iω m −ε is related to the spectral function A p (k, ε) .Finally, using the Matsubara frequency summation and Eq. ( 9), the transport coefficient ϒ mn obtained from the following equation: . The temperature dependence of the electrical conductivity σ (T) is proportional to 11 and is obtained by 66 : By using the obtained equations for the electrical and thermal conductivity, the temperature dependence of the Lorenz number and thermoelectric figure-of-merit can be defined in terms of the transport coefficients ϒ ij , as 66 : The heat capacity C(T) is the response of the total internal energy U ∂T ] and using the DOS spectrum, the electronic heat capacity C(T) in terms of the temperature, it has been defined as 67 :

Electronic properties
In this study, three types of doped h-BN structures have been selected for investigation: carbon dopant substitution on boron site [C B ], carbon dopant substitution on nitrogen site [C N ] and carbon dopants substitution on boron and nitrogen sites [C BN ].The electronic structure of doped h-BN with different impurity concentrations is shown in the Fig. 2 for N4 structure [supercell with four unit cells] with C B and C N impurity types.The electronic band structure was investigated in the symmetry path Γ-M-K-Γ.
A pure monolayer of BN is a wide band gap semiconductor.When it is doped with a carbon atom, a single subband is created within the band gap region.In the absence of a magnetic field (B T = 0), N 4 -doped structure remains as direct band gap semiconductor and exhibits several valence and conduction asymmetry paraboliclike subbands on both sides of the Fermi level.For the C B doped structure, the subband impurity above the Fermi level, shows flat dispersion in terms of wave vector k around the Γ point, especially in the MK direction (Fig. 2a1) and this feature is also occurs for the C N doped structure.For the C N doped structure, the impurity subband is located below the Fermi level in the valence band region and is flatness in terms of wave vector point around the K point (Fig. 2b1).In the case of C B -doped structures, the impurity subband above the Fermi level shows a flat dispersion in terms of wave vector k around the Dirac K point, especially in the MK direction (see Fig. 2a1).This feature is also observed in C N -doped structures and a similar flat impurity subband is present below the Fermi level around the K point in the valence band for C N doping (Fig. 2b1).In the presence of a magnetic field, the Hamiltonian becomes spin-dependent, which leads to spin splitting of the subbands.Each valence and conduction subband splits into two separated bands, one for spin up and one for spin down.With increasing magnetic field strength, further modifications are induced, including the increasing of band edges and the merging and crossing of subbands.Another interesting and important feature is the band gap reduction of doped BN under magnetic field, attributed to the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) shifting closer together.This band gap reduction becomes more prominent with increasing magnetic field strength.These behaviors are not limited to h-BN, but have also been observed in other nanostructures, such as boron and nitrogen doped silicon carbon nanotubes and carbon nanotubes [68][69][70] .
To investigate the effects of double carbon doping on the electronic properties of h-BN with different unit cells, the band structure of the N 4 structure with C BN impurities is shown in Fig. 3.In the absence of a perpendicular electric field, this double-doped structure retains the direct band gap at the Dirac point like pristine BN.The two carbon impurities introduce separate flat bands in the valence and conduction bands (Fig. 3a1).Without a magnetic field, the band gap for C BN -doped structure is smaller than that the same structures with one carbon impurity.This demonstrates that the band gap size depends on both the type and concentration of dopants.The application and enhancement of a magnetic field induces a greater band gap reduction in N 4 -C BN doped structures compared to those with just a single carbon impurity.Figure 3b shows the agreement between our tight binding results and DFT results for the band structure of CBN-doped h-BN.Good agreement is observed especially for the impurity subbands on both sides of the Fermi level, validating our tight binding approach.
Figure 4 shows effects of the magnetic field on the DOS spectra for doped h-BN with different impurity types and concentration.Without the magnetic field, the wide band gap in the n 5 -C B structure results in zero DOS intensity within |E|< 2 eV, because there are no available states for electrons in this energy range.The number of DOS peaks in the conduction region exceeds the valence region, attributable to the significant impact of the C dopant on the conduction bands near the Fermi level, regardless of magnetic field.The DOS spectrum for the n 5 -C B structure exhibits significant distinct peaks with high intensity in energies around + 2 eV which arise from the dopant impurity sublevel in the band structure.When a magnetic field is applied, the first conduction peak moves toward the band gap region with decreasing intensity which leads to the band gap reduction in this structure.The DOS of n 5 -C B structure, also has more and significant peaks in the positive energy region in the presence of a magnetic field.This is because the magnetic field enhances the effect of the dopant impurity sublevel on the conduction bands.In contrast, the DOS in the negative energy region has a single peak, which is due to the absence of the dopant impurity sublevel in this region.Figure 4b presents the influence of an applied magnetic field on the DOS for h-BN doped with C N impurity types.In the absence of a field, negligible DOS intensity is observed across a wide energy range around the Fermi level.For the C N dopant type, more DOS peaks appears in the valence region, compared to the conduction region.This asymmetry in C N doped structure, arises due to the significant influence of the carbon dopant on the valence band edges.For n 5 -C N , distinct high-intensity DOS peak appears around − 2 eV originating from the dopant impurity subband.Under an applied magnetic field, this impurity peak shifts toward the Fermi level, which leads to reduction of the magnetic field-induced band gap.As depicted in Fig. 4c, the presence of dual carbon dopants introduces two distinct peaks on either side of the Fermi level.This arises as the two dopants substantially influence both the valence and conduction bands.Under an applied magnetic field, the two first peaks nearest to the Fermi level, shift closer together and this leads to the band gap reduction.
It can be concluded that the number, position and intensity of the peaks strongly modified with dopant type and concentration.Also, the magnetic field creates many additional peaks inside the band gap region by splitting the valence and conduction subbands and moving them toward the Fermi level.

Thermal properties
Thermal properties of materials are affected by electronic and phononic contributions.This study focuses on the electronic contribution under conditions of high electron concentration and short phonon mean free path and the phononic contribution is not considered here 71 .
The temperature dependence of the heat capacity of h-BN with the carbon impurity and magnetic field has been investigated using the Eq. 13.For this purpose, the n4-doped structure with C B , C N and C BN dopant types are selected.In the T < 2000 K, the C(T, B T = 0) for n4-CB doped structure exhibits near to zero intensity (red line in Fig. 5a) due to its wide band gap which acts as a barrier potential for the charge transition.Below 2000 K, the charge carriers lack sufficient thermal energy for transfer to higher levels.The magnetic field decreases the band gap and significantly alters the DOS peak intensities near the Fermi level.Due to these modifications, the density of charge carrier increases and with lower thermal energy, more charge carriers can be transfer to the higher levels.So, in non-zero magnetic field B T ≠ 0, the heat capacity of n 4 -C B case becomes non zero below 2000 K.As shown in the Fig. 5a, the C(T, B T = 1) remains zero below 1000 K and becomes non zero above this temperature region.The increasing rate of heat capacity with temperature depends on the magnetic field strength.At higher fields of B T = 1.4 and 1.6, the heat capacity becomes non zero and rises when temperature exceeds 500 K and 300 K, respectively.Additionally, at B T = 1.8, C is nonzero across all temperatures below 2000 K and grows linearly with increasing temperature.
Figure 5b compares the effects of dopant impurity on the heat capacity C(T) for different doped structures (n 3 , n 4 , n 5 supercells) under a constant magnetic field.At B T = 1, all structures exhibit identical C(T) intensity below 1500 K. Above 1500 K, n 5 has the largest C(T) while n3 has the smallest.This trend also occurs at higher magnetic fields of B T = 1.6 and 1.8, where n 5 and n 3 display the largest and smallest C(T) respectively for T > 500 K. Notably, the increasing rate of C(T) with temperature is enhanced by increasing magnetic field.Furthermore, at higher fields, the difference in C(T) between structures increases.In summary, besides dependence on magnetic field, C(T) also strongly depends on the impurity concentration, with lower concentrations exhibiting larger C(T).
The temperature dependence of the specific C(T) for the n 4 -C BN doped structure containing two carbon impurity atoms is presented in Fig. 6a.In the absence of an external B T , the C(T) intensity remains close to zero below 2000 K.This is attributed to the wide band gap of the n 4 -C BN doped structure, which inhibits thermal excitation of charge carriers to higher energy levels.The C(T) as a function of temperature for the n 4 -C BN doped structure exhibits a strong dependence on the applied external magnetic field B T .In B T = 0, C(T) remains negligible up to 2000 K due to the wide band gap of the n 4 -C BN which prevents significant thermal excitation of charge carriers.However, the application of a magnetic field induces a reduction in the band gap via the Zeeman are approximately equal to each other, and both are lower than the C(T) for the C BN doped structure.However, at the higher field strength of B T = 1.8, the differences in C(T) between the dopant types decrease as the temperature rises above 1000 K (Fig. 6d).
Figure 7 shows the electronic contribution to electrical conductivity σ(T) versus temperature for C B -doped structures with different doping concentrations and magnetic fields.The temperature dependence of the σ(T) depends on different parameters such as the band gap of doped structure, density of excited charge carriers and their mobility.Without an external B T , the wide band gap about 3.6 eV for the n 4 -C B doped structure prevents charge carrier excitation, giving zero σ(T) below 2000 K as seen in Fig. 7a.The application of an external magnetic field causes Zeeman splitting of the electronic energy levels in the doped h-BN, which reduces the band gap.The band gap decreases to 0.76 eV and 0.32 eV at magnetic field strengths of B T = 1.4 and 1.6, respectively.The band gap then vanishes completely at a magnetic field of BT = 1.8.The smaller band gap enables increased thermal excitation of electrons from the valence band to the conduction band at lower temperatures.The greater number of thermally excited charge carriers leads to an increase in σ(T).Consequently, compared to the absence of the  www.nature.com/scientificreports/as a result, the σ(T) stays negligible or zero.With increasing applied magnetic fields of B T ≥ 1.6, the σ(T) becomes nonzero above 500 K, respectively.This is because the band gap reduction caused by this magnetic field strength is still large enough to prevent thermal excitation of carriers below 500 K.However, at the higher magnetic field of B T = 1.8, the band gap has been sufficiently reduced so that the σ(T) reaches its maximum intensity and becomes nonzero at temperatures above 0 K.It can be concluded that the C B doped structure exhibits a larger band gap when is influenced with weaker BT, so, its σ(T) increases when a stronger B T is applied.Figure 7c and d show the σ(T) for C B -doped h-BN structures with different impurity concentrations n 4 and n 5 , under various magnetic fields.At B T = 1, the σ(T) for both structures is zero below 1200 K and approximately equal.When the magnetic field is increased to B T = 1.4, the σ(T) remains zero below 500 K but significantly increases above this temperature, with a higher intensity for the n 5 doped structure.This increase in electrical conductivity is due to the magnetic field's effect on the band gap, as the gap decreases with increasing B T .Note that, the rate of band gap reduction with respect to B T is greater for the n 5 structure compared to the n 4 structure.At stronger magnetic fields of B T = 1.6 and 1.8, the difference between σ(T) of n 4 and n 5 increases, with n 4 having a lower σ(T) intensity.This is due to n4 having a larger band gap than n5 at these field strengths.These findings also can be explained by the impurity concentration.The doped h-BN structure with lower impurity concentration (n 5 ) allows electrons and holes to move more freely with less scattering.This enables more efficient heat conduction and leads to higher electrical conductivity, when compared to the structure with higher impurity concentration (n 4 ) which causes more scattering.
The results in Fig. 8 indicate that the electrical conductivity of doped h-BN exhibits a strong dependence on several external factors, namely temperature, dopant type and concentration, and magnitude of the applied magnetic field.By engineering these external factors, significant changes can be induced in the electrical conductivity.The tunable electrical conductivity of doped h-BN originates from the changes in the electronic band structure caused by the substitution of carbon atoms in h-BN with boron or nitrogen dopants.The impurity atoms introduce new energy levels within the band gap that are sensitive to both doping concentration and applied magnetic field strength.Tuning the external factors leads to changes in the band structure, specifically decreasing the size of the band gap.The modifications become more pronounced as the magnetic field increases.Consequently, the narrowed band gap requires less energy for thermal excitation of charge carriers across the gap, resulting in higher electrical conductivity.For all selected doped structures, the electrical conductivity is zero in the finite temperature region indicated by TZ.The TZ region depends on the impurity concentration and magnetic field strength.At a magnetic field of B T = 1.4, the σ(T) remains zero below temperatures of T Z = 700 K and 400 K for the C B and C BN doped structures, respectively (Fig. 8a).Above these temperatures, the σ(T) increases from zero for both doped structures as the temperature further increases because the higher temperatures provide the required thermal energy to excite a greater number of charge carriers to higher energy levels.At B T = 1, the σ(T) of the n 4 -C B doped structure is smaller than the n 4 -C BN doped structure, and this relationship is unchanged as B T increases to 1.4.This can be explained by the smaller band gap in the C BN -doped structure compared to the C B -doped structure, which requires less thermal energy to excite charge carriers to higher levels.Figure 8b shows the behavior of σ(T) in the presence of stronger magnetic fields B T = 1.6 and 1.8, for the C B and C BN doped structures with the same n 4 supercell.At B T = 1.6, the σ(T) of the CB-doped structure is zero below T z < 500 K [due to the non-zero band gap] and increases above 500 K.This differs from the behavior of the C BN -doped structure at the same magnetic field of B T = 1.6, which has a non-zero σ(T) even below 2000 K.This difference can be attributed to the disappearance of the band gap in the n 4 -C BN structure when B T = 1.6.However, the behavior of both structures at B T = 1.8 reveals an interesting difference.While the σ(T) is non-zero below 2000 K for both, the C B structure displays a linearly increasing σ(T) pattern, whereas the C BN structure exhibits a decreasing σ(T) above 500 K.The decreasing σ(T) pattern for C BN at the high magnetic field of B T = 1.8 can be explained by scattering among charge carriers.The C BN band gap becomes very narrow at B T = 1.8.As temperature increases, more charge carriers gain enough energy to occupy higher energy levels, which rapidly increases the population of excited carriers.This leads to increased scattering interactions between charge carriers at higher energy levels as temperature increases.These more scattering interactions between carriers lead to the reduced σ(T) exhibited by the C BN -doped structure.It can be concluded that (i) C BN doping gives higher conductivity than C B doping due to a smaller band gap and (ii) high densities of scattered carriers can decrease conductivity as seen for C BN -doped h-B N at high magnetic fields.
The temperature behavior of the electronic figure of merit ZT(T) dependent on dopant type and concentration is investigated in Fig. 9 for the N 3 , N 4 , and N5 doped structures with C B impurity.In the absence of a magnetic field (red lines in Fig. 9), ZT(T) has negligible values below approximately 1000-1200 K, then sharply increases above this temperature range.This pattern occurs independently of impurity concentration for all selected cases.Also, similar temperature-dependent increases in ZT(T) with temperature have also been reported for Silicene and Germanene 72,73 and MoTe2 74 .Applying a magnetic field to each structure causes the ZT(T) increase to begin at lower temperatures.For example, for the n 3 -C B doped structure, the ZT(T) becomes non zero and increases above 500 K and 260 K when magnetic field becomes B T = 1 and 1.4, respectively.The magnetic field B T has another significant effect on ZT(T) intensity, creating larger ZT(T) values at higher B T strengths relative to weaker B T .As the temperature increases to 2000 K, each chosen structure displays a unique ZT(T) behavior.As temperature increases towards 2000 K, the ZT(T) for n 3 -C B converges to the same value and the difference in ZT(T) for varying magnetic field strengths decreases.For the n 4 -C B doped structure, the difference in ZT(T) between varying magnetic field strengths increases slightly, with stronger B T exhibiting higher ZT(T) intensity (Fig. 9b).In the presence of non-zero magnetic fields, the ZT(T) of the n 5 -C B doped structure displays a peak at temperature T M (Fig. 9c) and As B T increases, this peak position in the n 5 -C B ZT(T) shifts to lower temperatures.Notably, comparison of the ZT(T) intensity for the n 3 , n 4 , and n 5 doped structures with the same impurity type shows that n5 exhibits the highest intensity compared to the other cases.Based on the above results, it can be concluded that the ZT(T) is higher for structures with lower impurity concentrations and increases as the impurity concentration is reduced.Figure 9d shows the effects of the dopant type on temperature dependence of ZT(T) for n 4 -C B and n 4 -C N doping at various magnetic fields.The results demonstrate that at a given magnetic field strength, the C B doped structure exhibits a higher ZT(T) compared to the C N doped structure.
Figure 10 displays the influences of magnetic field, impurity type, and concentration on the temperaturedependent power factor PF(T) for the chosen structures.For the N 4 structure with C B -type doping, the power factor PF(BT = 0) remains near zero below 2000 K.When the magnetic field B T = 1 is applied, the PF(T) rises above 1500 K, and at B T = 1.8 the increasing rate is substantially enhanced with temperature (Fig. 10a).Figure 10b compares the PF(T) for n 4 and n 5 doped structures with C B impurities at varying B T .At B T = 1.2, the PF(T) is zero below 1000 K and rises above 1000 K for both structures, with greater intensity for the n5 structure.As magnetic field increases to B T = 1.6, the PF(T) for n 5 grows at a faster rate compared to the n 4 structure and their differences more increased as temperature reaches 2000 K.The results indicate that reducing the impurity concentration has a greater impact on PF(T) compared to increasing the magnetic field strength B T , as evidenced by Fig. 10b, where the PF(T) at BT = 1.6 for n 5 -C B doping exceeds the PF(T) at B T = 1.8 for n 4 -C B doping.In summary, for a given impurity type, the PF(T) can be increased by both raising the magnetic field strength and lowering the impurity concentration.However, reducing the impurity level has a greater enhancing effect on PF(T) compared to increasing the magnetic field.As shown in the Fig. 10c and d, the intensity of PF(T) with the C B impurity type is larger than that corresponding doped structure with the C N impurity type and this behavior is independent of the magnetic field strength and impurity concentration.

Conclusion and outlooks
Optimizing the thermoelectric performance of carbon-doped h-BN monolayers through tuning the doping concentration, magnetic field strength, and impurity type is investigated through tight binding model, using Green's function approach and Kubo formalism.Through determination of accurate tight-binding parameters, excellent agreement was obtained between the electronic properties of the tight-binding model and corresponding Density Functional Theory (DFT) calculations for doped h-BN structures across a range of impurity types and concentrations.The effects of external factors such as carbon doping type and concentration as well as applied magnetic field strength were systematically studied through their influence on key thermoelectric properties including the electronic density of states, electrical conductivity, heat capacity, electronic figure of merit, and power factor.The electronic band structure and density of states of monolayer hexagonal boron nitride (h-BN) are significantly modified by carbon doping.Carbon dopants induce subband splitting and merging in the electronic structure, altering the number, position, and magnitude of peaks in the density of states.Under an applied magnetic field, additional peaks appear in the band gap region of C-doped h-BN, originating from sublevel splitting which leads to a reduction of the band gap in carbon-doped h-BN.
These modifications to the electronic structure directly influence the electrical conductance and thermoelectric properties of h-BN as follows: • Double carbon doping improves electrical conductivity compared to single doping, but strong carrier scat- tering reduces it at high magnetic fields.• Heat capacity varies markedly with dopant concentration, displaying higher values at lower dopant levels.
• At a given magnetic field, the CB doped structure exhibits a higher electronic figure of merit ZT(T) than the CN doped structure, with ZT(T) increasing as impurity concentration decreases.• For a given dopant type, the power factor PF(T) can be raised by increasing magnetic field and lowering impurity concentration.
The ability to tune the electronic structure and band gap of h-BN through carbon doping and applied magnetic fields opens up new possibilities for engineered optical and electronic properties.This work shows routes to enhance h-BN for advanced optoelectronic and power generation applications using band gap modification through doping combined with electronic structure changes induced by magnetic field.

R α i Figure 1 .
Figure 1.Schematic picture of the atomic structure for the N 4 -doped h-BN with (a) C B , (b) C N and (c) C BN .

Figure 3 .
Figure 3.The tight binding results for the band structure of (a1-a3) n4-CBN doped h-BN structures in the presence of the magnetic fields Π = 0, 1, 2, respectively.(b) The tight binding (blue lines) and DFT (red lines) band structures.Good agreement is observed especially for the impurity subbands on both sides of the Fermi level, validating our tight binding approach.

Figure 4 .Figure 5 .
Figure 4.The DOS of the h-BN structure with (a) n5-CB, (b) n5-CN and (c) n4-CBN dopant types in the presence of the rescaled magnetic field B T .

Figure 6 .
Figure 6.(a) Heat capacity [in an arbitrary unit] as a function of temperature for n4-CBN doped h-BN monolayer at various applied rescaled magnetic fields.(b-d) Comparison of heat capacity versus temperature for n4 doping concentration with CB, CN, and CBN dopant types under rescaled magnetic fields of 1, 1.4, and 1.8, respectively.The results demonstrate tunable heat capacity through the magnetic field, dependent on both dopant type and concentration.

Figure 7 .
Figure 7. (a, b) Temperature dependence of electrical conductivity [in an arbitrary unit] for n4 CB-doped h-BN monolayer under different applied rescaled magnetic field strengths.(c, d) Comparison of electrical conductivity as a function of temperature for n4 and n5 CB doped h-BN at various rescaled magnetic field strengths.The results demonstrate tunable electrical conductivity through the magnetic field, dependent on both dopant concentration and magnetic field intensity.

Figure 8 .
Figure 8.Comparison of electrical conductivity [in an arbitrary unit] versus temperature for n4 doping concentration with CB and CBN dopant types under different rescaled magnetic fields.

Figure 9 .
Figure 9. Tuning thermoelectric figure of merit ZT(T) in carbon doped h-BN through magnetic fields.(a-c) Temperature-dependent ZT(T) for CB doped h-BN monolayers with n3, n4, and n5 dopant concentrations under applied rescaled magnetic fields.(d) Comparison of ZT(T) as a function of temperature for n4 doped h-BN with CB and CN dopant types at various rescaled magnetic field strengths.

Figure 10 .
Figure 10.(a, b) Temperature-dependent of power factor PF(T) [in an arbitrary unit] for CB doped h-BN monolayers with n4, and n5 dopant concentrations under applied rescaled magnetic fields.(c, d) Comparison of PF(T) as a function of temperature for n4 doped h-BN with CB and CN dopant types at various rescaled magnetic field strengths.